Chapter6_4 - ) that we can ever know about the particle it...

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PHY3063 R. D. Field Department of Physics Chapter6_4.doc University of Florida Expectation Values and Differential Operators (2) Dynamical Variables become Differential Operators: x i p op x = h ) ( 2 2 2 2 ) ( x p op x = h t i E op = h Expectation Values of Dynamical Quantities: The average momentum is given by Ψ Ψ = Ψ Ψ >= < dx x t x t x i dx t x p t x p op x x ) , ( ) , ( ) , ( ) )( , ( * * h Ψ Ψ = Ψ Ψ >= < dx x t x t x dx t x p t x p op x x 2 2 * 2 2 * 2 ) , ( ) , ( ) , ( ) )( , ( h and the average energy is Ψ Ψ = Ψ Ψ >= < dx t t x t x i dx t x E t x E op ) , ( ) , ( ) , ( ) , ( * * h and in general Ψ Ψ >= < dx t x t x i x f t x t p x f op x ) , ( ) , , ( ) , ( ) , , ( * h , where I have assumed that 1 ) , ( ) , ( * = Ψ Ψ dx t x t x . The wave function Ψ (x,t) contains all the information ( consistent with the uncertainty principle
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Unformatted text preview: ) that we can ever know about the particle it represents. Position Space Operators: An operator acting on a function maps it onto another function. The operators we have considered so far are position space operators because that operate on the set of all square integrable position space wave functions, (x,t) , producing another square integrable position space wave function as follows: ) , ( ) , ( t x t x O op = In position space: (x) op = x and (x 2 ) op = x 2 , etc. Functions of position x and time t...
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